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Electronic Structure of Condensed MatterDavid M. CeperleyRichard M. Martin

• Electron Gas - Wigner Crystal • Hydrogen• Real Materials – Si, Al surface, molecules, …• Quantum Dots• Theory of insulators – polarization and

localization – metal-insulator transitions

• Goal: to solve THE many-body electronic structure problem in condensed matter

∑∑≠

+∇−=ji ji,

jiN

i

2i

i

2

r ee

1 m1

2ˆ

e

εH

rh NucleiElectrons

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Electronic Structure of Condensed MatterDavid M. CeperleyRichard M. Martin

• Goal: to solve THE many-body electronic structure problem in condensed matter

∑∑≠

+∇−=ji ji,

jiN

i

2i

i

2

r ee

1 m1

2ˆ

e

εH

rh NucleiElectrons

• Special features of our work– Combine expertise and concepts of:

Quantum Monte Carlo (QMC)Density Functional Theory (DFT)Analytic many-body theory

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Theme of Recent WorkAdvantage of Monte Carlo:Averages are almost free.

Suppose we have an extra parameter “q” to sum over.

• Deterministic calculation: CPU time is multiplied by M.

• Monte Carlo calculation: almost no slow down since the calculation is just one more variable to average over.

3N → 3N + 1

• Especially adapted for parallel computers: Simply run M calculations on M separate processors for different values of q: they all serve to reduce the error bar.

The only slow down occurs comes from “start up” costs: e. g. Metropolis warm-up

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1( ) ( ; )

M

iMi

E s E s q=

= ∑

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Advantage of Monte Carlo:

Averages are almost free.

Examples of averages:1. k-point sampling (integrate over Brillouin zone of

supercell). Twist averaged boundary conditions converge much faster than periodic boundary conditions for metals. (M k-points)

2. Path Integrals for ions (particularly for protons or light ions) (M time slices to average over.)

3. Interactions of electrons in real complex quantum devices with different dielectric media and metal gates (M steps in the Dyson Green’s function equation)

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Recent Progress on Simulations of the Low Density Electron Gas

• New Methods have led to progress in predicting the phase diagram– Path Integral Monte Carlo – Better wavefunctions (Kwon*, Holzmann)– Twisted Boundary conditions (Lin*, Zhang*)

• Wigner Crystal melting (Jones, Candido)• Spin polarization (Lin*, Zong*)

*Supported by NSF Grant

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( )i j ij

eV R background

r<

= +∑

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3D electron gas Brief History of Ferromagnetism

What is polarization state of fermi liquid at low density?

• Bloch 1929 polarization from exchange interaction:– rs > 5.4 3D– rs > 2.0 2D

• Stoner 1939: include electron screening: contact interaction

• Herring 1960• Ceperley-Alder 1980: rs >20• Young-Fisk experiment on doped CaB6 1999 • Ortiz-Balone 1999 : ferromagnetism of e gas. • Our new work 2001: “twist averaged boundary

conditions”

↓↑

↓↑

+−

=ζNNNN

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Twist averaged boundary conditions(or k-point sampling)

• Use a phase or twist as r → r+L. (Bloch boundary condition)

• For fermi liquid, one can eliminate single particle shell effects by averaging over the twist angle:

• Gas: Momentum distribution changes from a lattice to a Fermi sea

• No more shell effects!

kxkx

( ) ( )ir L e rθψ ψ+ =

3

1(2 )

O d Oθ θθ ψ ψπ

= ∫

2

ikre

kL n

ϕπ θ

== +

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Polarization with backflow

• Twist averaging allows calculation with small systems and high quality wavefunctions

• Polarization is sensitive to wavefunction– Jastrow

wavefunctions favor the ferromagnetic phase.

– Improved Backflow 3-body wavefunctions are more paramagnetic

• General rule: any approximation favors ferromagnetism

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Coupled Ionic-Electronic Simulations

(CIEMC)

• Much progress in recent years with “ab initio” molecular dynamics simulations.

• However density functional theory is not always accurate enough.

• Is it possible to go to the next level of accuracy in description of the electronic structure?

– Hard sphere MD/MC ~1953 – Empirical potentials (e.g. Lennard-Jones) ~1960 – Local density functional theory

Car-Parrinello MD simulations ~1985– Quantum Monte Carlo

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Phase Diagram of Hydrogen

• Full PIMC works well at higher temperatures (T > EF /10)

• We need a method for intermediate temperatures

• DMC works well at T=0

(but time scale problem Mp/Me)

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A single CIEMC step

•Move ions from S to S*•Pre-reject moves based on empirical potential•Re-optimize trial function ΨT (R|S*;a) w.r.t. “a”.•Sample electron from P(R,S,S*)•Determine energy difference.•Use penalty method to accept or reject

electrons

ions

R

S ⇒ S*

Problem – “Noise” from sampling electrons

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The Penalty methodJ. Chem. Phys. 110, 9812(1998).

• Assume estimated energy difference ∆e is normally distributed* with variance σ2 and the correct mean.

< ∆e > = ∆E< [∆e- ∆E]2 > = σ2

*results from averaging according to central limit theorem if σ<∞

• Acceptance probability: a(x,σ) = min [ 1, exp(-x- σ2/2)]

• Simply add an additional “penalty” to the energy difference!• The extra factor σ2/2 is the penalty • This exactly satisfies detailed balance and hence converges to

the Boltzmann distribution independent of noise level.

Protons treated classically at present

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CEIMC for Hydrogen • Solved major problem: Improved “backflow”

wavefunction involving electrons and protons allows reoptimization during the CEIMC run.

• No parameters, no LDA calculation needed at each step.An LDA/Jastrow trial function is too slow!

• Especially appropriate for liquid metallic hydrogen and disordered systems.

• The result is a very fast CEIMC code (faster (?) than Car-Parrinello-MD for metallic hydrogen) which treats electrons and protons by accurate QMC simulations.

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- Compare some results to coupled cluster Compare some results to coupled cluster methods (group of R. Bartlett, U. of Florida)methods (group of R. Bartlett, U. of Florida)

Correlated Electrons in Quantum DotsCorrelated Electrons in Quantum DotsSpin States, Charge GapsSpin States, Charge Gaps

Quantum Monte Carlo StudiesQuantum Monte Carlo Studies

Tim Tim WilkensWilkensHoonHoon KimKim

Curry Taylor (undergrad)Curry Taylor (undergrad)Richard M. MartinRichard M. MartinDavid M. David M. CeperleyCeperley

Philippe Philippe MatagneMatagne, J. P. , J. P. LeburtonLeburton(Dept. of (Dept. of ElecticalElectical Engineering)Engineering)

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- Dots created by layered materials, applied potentialsDots created by layered materials, applied potentials

- Effective mass approximation validEffective mass approximation valid

-- IdealizedIdealized and and realisticrealistic dots dots

Electrons in Quantum DotsElectrons in Quantum Dots

What type of dots? What type of dots?

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Electrons in Quantum DotsElectrons in Quantum Dots

“Squashed” pancake shaped dot “Squashed” pancake shaped dot

Often idealized as 2Often idealized as 2--dimensionaldimensionalOur work: full 3Our work: full 3--dimensional interacting electron problem dimensional interacting electron problem

First QMC calculation for “real” system, m*( r ) , …First QMC calculation for “real” system, m*( r ) , …

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FEM GRIDFEM GRID

• only center portion of FEMFEM grid utilized:

XX--YY: : 51 51 ptspts ZZ: 33 pts: 33 pts{{{{{{{{ −−−−−−−−750750:750 750 AA°°°°°°°°}}}}}}}} {{{{{{{{ −−−−−−−−120120:120 120 AA°°°°°°°°}}}}}}}}

• vertical confinement ratio vertical confinement ratio ~ 33

• Hund’sHund’s RuleRule satisfied for allCases of NNee

• LDALDA and DMCDMC differ only for1111 electrons

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MC simulation of full device?• Poster – Das, Martin, Kalos , Ceperley • Green’s function Monte Carlo for Coulomb

problem in complex geometry with gates, ….. • Example of Advantage of MC: 3N → 3N + 3• In progress

for Coulomb Potentials

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What is an insulator? Brief History: Polarization & Localization

• Classical theory of E&M in matter:Free charge in metals - real currentsBound charge in insulators – polarization currents

• Kohn 1964 “Theory of the insulating state”– Localization– Due to Pauli principle, disorder, interactions

• Martin 1974 Problem in extended matterPolarization cannot be determined from the bulk density

• Kingsmith & Vanderbilt 1993 Polarization determined by Berry’s phase

• Ortiz & Martin 1994, 97Many-Body Berry’s phase formulationDensity-polarization theory

• Souza, Wilkens & Martin - 2001 New formulation – rigorous definition of localization length

NSF

Lesson from statistics: Generating Functions

• Characteristic Function:C(αααα) = ∫ exp(- i αααα X) p(X) dX

• Moments:

< X X … >n = in (d/d αααα)n C(α α α α )|αααα=0

• Cumulants:

< X X … >cn = in (d/d αααα)n ln C(α α α α )|αααα=0

• First Cumulant: < X >c = < X >

• Advantage: Higher cumulants independent of origin, e.g.,

< X2 >c = - (d/d αααα)2 ln C(α α α α )|αααα=0

= < X2 > - < X >2 = < ∆X2 >

Moments of Polarization – Quantum Fluctuations

• Souza (2001) – extension to infinite systemsC(αααα) = <<<< ΨΨΨΨ ||||exp(- i αααα X) |||| Ψ Ψ Ψ Ψ >>>>

• Cumulants of P:

< P P … >cn = (e n /V) in (d/d αααα)n ln C(α α α α )|αααα=0

• < ∆∆∆∆P2 >c defines localization length

• Fluctuation-dissipation theorem for finite system:< ∆P2 > = (e 2 /V) < ∆X2 >

= (h/2 π2) ∫∫∫∫ Re σ (ω) d ω / ω

• And in the limit of large V< ∆P2 > = finite for insulators< ∆P2 > = infinite for metals

Measurable

Twisted boundary conditions & observables

• Souza, et al., PRB 62, 1666( 2000) -- Extends famous work of Kohn - 1964 - “Theory of the insulating state”

• Useful for insulatorsApproach to Metal-Insulator transition from insulating side

QMC calculation of Polarization, Localzation

• Ionic Hubbard Model in 1D – Wilkens - 2001• Spontaneous symmetry breaking at transition

Symmetric lattice

Gap vanishesζ diverges

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Density Functional Calculations

• Poster – Igor Vasiliev, Martin (Chelikowsky)• Time dependent DFT for excitations• Effect of Oxygen on Optical properties of

Silicon Clusters

Example

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Density Functional Calculations

• Poster – Romero, Kim, Ordejon, Mozos, Martin• FET’s in C60?• Nature of surface states - Superconductivity

Example: Added Holes

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Electronic Structure of Condensed MatterSummary

• Fundamental Advances toward the goal of solving THE many-body electronic structure problem in condensed matter

• Advantages of QMC– Many body method– Can use statistical sampling to advantage!– Electron gas, quantum dots, hydrogen, real

materials• Fundamental Advances in theory of insulators

– Polarization & Localization• Applications using DFT

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Electronic Structure of Condensed MatterFuture

• QMC methods that:– Solve the real many-body problem– Compete with DFT in real applications

• New DFT approaches:– Develop Density-Polarization theory– Excitations

• Applications – Idealized and real systems - QMC and DFT

http://archive.ncsa.uiuc.edu/Apps/CMP/cmp-homepage.htmlhttp://www.physics.uiuc.edu/research/ElectronicStructure/index.html